Modelling decontamination of two-dimensional spills Oliver - PowerPoint PPT Presentation

University of Oxford Modelling decontamination of two-dimensional spills Oliver Whitehead Mathematical Institute Supervisors: Chris Breward, Ian Griffiths (Oxford), Ross Heatlie- Branson (DEFRA) and with Ellen Luckins EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Porous Mathematical Institute Medium, (Concrete) EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Agent Porous Mathematical Institute Medium, (Concrete) EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Porous Mathematical Institute Medium, (Concrete) EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Porous Mathematical Institute Medium, (Concrete) ‘ The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium’ by Liu, Zheng, and Stone EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Porous Mathematical Institute Medium, (Concrete) ‘ The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium’ by Liu, Zheng, and Stone EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Cleanser Porous Mathematical Institute Medium, (Concrete) ‘ The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium’ by Liu, Zheng, and Stone EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Cleanser Porous Mathematical Institute Medium, (Concrete) ‘ The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium’ by Liu, Zheng, and Stone EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Modelling Assumptions University of Oxford • Agent and cleanser are immiscible • The agent is neat – not in solution • The reaction occurs only at the interface and is: • Agent (𝑚) + Cleanser (𝑏𝑟)  Product • Product insoluble in the agent • Product in solution does not affect cleanser in solution • Assume that there is no fluid flow Mathematical Institute • Below the agent is unsaturated • No evaporation of cleanser or agent • The layers of cleanser and agent are thin EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Porous Medium, Mathematical Institute (Concrete) EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford Non-porous Medium (Air) Length Height Porous Medium, Mathematical Institute (Concrete) EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Physical Set-up University of Oxford 𝐼𝑓𝑗𝑕ℎ𝑢 𝑀𝑓𝑜𝑕𝑢ℎ ≈ 0.1 ≪ 1 Non-porous Medium (Air) Length Height Porous Medium, Mathematical Institute (Concrete) EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford Mathematical Institute 𝑍 𝑌 EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 𝜖𝑑 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 Mathematical Institute 𝑍 A 𝑌 EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 𝜖𝑑 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 𝜖𝐷 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝐷 B. Mathematical Institute 𝑍 A 𝑌 B EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 𝜖𝑑 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 𝜖𝐷 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝐷 B. C. Neat Agent Mathematical Institute 𝑍 A 𝑌 B C EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 𝜖𝑑 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 𝜖𝐷 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝐷 B. C. Neat Agent D. Unsaturated medium Mathematical Institute 𝑍 A 𝑌 B C D EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 1. No flux, 𝜖𝑑 𝒐 𝒖𝒑𝒒 · 𝛼𝑑 = 0 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 𝜖𝐷 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝐷 B. C. Neat Agent D. Unsaturated medium 𝒐 𝒖𝒑𝒒 Mathematical Institute 1 𝑍 𝑍 = ℎ(𝑌) A 𝑌 B C D EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 1. No flux, 𝜖𝑑 𝒐 𝒖𝒑𝒒 · 𝛼𝑑 = 0 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 2. Continuity, 𝜖𝐷 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝐷 B. 𝑑 = ϕ𝐷, 𝑑 𝑍 = ϕ𝐷 𝑍 C. Neat Agent D. Unsaturated medium 𝒐 𝒖𝒑𝒒 Mathematical Institute 1 𝑍 𝑍 = ℎ(𝑌) A 2 𝑌 B C D EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 1. No flux, 𝜖𝑑 𝒐 𝒖𝒑𝒒 · 𝛼𝑑 = 0 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 2. Continuity, 𝜖𝐷 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝐷 B. 𝑑 = ϕ𝐷, 𝑑 𝑍 = ϕ𝐷 𝑍 C. Neat Agent 3. No flux, D. Unsaturated medium 𝐷 𝑍 = 0 𝒐 𝒖𝒑𝒒 Mathematical Institute 1 𝑍 𝑍 = ℎ(𝑌) A 2 3 𝑌 B C D EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 1. No flux, 𝜖𝑑 𝒐 𝒖𝒑𝒒 · 𝛼𝑑 = 0 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 2. Continuity, 𝜖𝐷 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝐷 B. 𝑑 = ϕ𝐷, 𝑑 𝑍 = ϕ𝐷 𝑍 C. Neat Agent 3. No flux, D. Unsaturated medium 𝐷 𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼 𝜖𝑢 + 𝐸 𝑑 (𝒐 𝑰 · 𝛼)𝐷 = −𝑙𝐷 , 𝜖𝐼 𝜖𝑢 = 𝑙χ𝐷 𝒐 𝒖𝒑𝒒 Mathematical Institute 1 𝑍 𝑍 = ℎ(𝑌) A 2 3 𝑌 𝑍 = −𝐼(𝑌, 𝑢) B C 4 D 𝒐 𝑰 EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling

Mathematical Model University of Oxford A. Diffusion of cleanser, 1. No flux, 𝜖𝑑 𝒐 𝒖𝒑𝒒 · 𝛼𝑑 = 0 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝑑 2. Continuity, 𝜖𝐷 𝜖𝑢 = 𝐸 𝑑 𝛼 2 𝐷 B. 𝑑 = ϕ𝐷, 𝑑 𝑍 = ϕ𝐷 𝑍 C. Neat Agent 3. No flux, D. Unsaturated medium 𝐷 𝑍 = 0 4. Chemical Reaction, 𝐷 𝜖𝐼 𝜖𝑢 + 𝐸 𝑑 (𝒐 𝑰 · 𝛼)𝐷 = −𝑙𝐷 , 𝜖𝐼 𝜖𝑢 = 𝑙χ𝐷 5. No flux, 𝒐 𝒖𝒑𝒒 𝒐 𝒄𝒑𝒖 · 𝛼𝑑 = 0 Mathematical Institute 1 𝑍 𝑍 = ℎ(𝑌) A 2 3 𝑌 𝑍 = −𝐼(𝑌, 𝑢) B C 4 D 𝒐 𝑰 𝒐 𝒄𝒑𝒖 5 𝑍 = −𝑒(𝑌) EPSRC RC Centre for Doctor oral al Training ning in Indust dustrially ially Focuse sed Mathema matical ical Modelling lling